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On finitely generated vector sublattices

Volume 245 / 2019

Lech Drewnowski, Witold Wnuk Studia Mathematica 245 (2019), 129-167 MSC: Primary 46B42, 28A05, 54H05. DOI: 10.4064/sm170524-23-12 Published online: 20 July 2018

Abstract

We investigate various questions concerning vector sublattices, $\operatorname {vlt} S$, generated by subsets $S$ of an Archimedean vector lattice $E$. We first prove a distributivity law: $\operatorname {vlt}(X;Y,Z)=\operatorname {vlt}(X,Y)+\nobreakspace {}\operatorname {vlt}(X,Z)$ if $X,Y,Z\subset E$ and $Y\perp Z$, and derive a number of its consequences. We next show that in a topological vector lattice the dimension of the sublattice generated by an analytic set is either $\le \aleph _0$ or $2^{\aleph _0}$, and that the same is true for sublattices generated by at most countable sets in arbitrary vector lattices. In a vector lattice, we characterize those sets that generate $n$-dimensional sublattices and prove that a finite set generates a finite-dimensional sublattice if so does each pair of its elements. We also show that in a uniformly complete vector lattice every principal ideal of infinite dimension contains pairs of positive elements generating $\aleph _0$- as well as $2^{\aleph _0}$-dimensional sublattices. The special case of lattices $C(K)$ is also treated in this respect. Moreover, for a compact set $K\subset \mathbb R^n$ with a nonempty interior, it is shown that the minimal number of functions in $C(K)$ or $C(K)_+$ generating a dense sublattice is $n+1$. We also prove that every (separable) Banach lattice $C(K)$ can be embedded in a discrete (separable) Banach lattice of the same type. Finally, we prove that in a discrete and $\sigma $-Dedekind complete separable $F$-lattice one can always find a pair of positive elements generating a dense sublattice, and we use that result to show that, in general, this is far from being true even in the case of discrete separable $C(K)$ lattices.

Authors

  • Lech DrewnowskiFaculty of Mathematics and Computer Science
    A. Mickiewicz University
    Umultowska 87
    61-614 Poznań, Poland
    e-mail
  • Witold WnukFaculty of Mathematics and Computer Science
    A. Mickiewicz University
    Umultowska 87
    61-614 Poznań, Poland
    e-mail

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